As you mentioned, you cannot use Query Store or Extended Events to monitor queries on read-only replicas but still can use sys.dm_exec_query_stats dynamic view, which can give you for example top queries by CPU consumption
SELECT TOP (25) MIN(query_stats.statement_text) AS [Statement Text],
SUM(query_stats.total_worker_time) / SUM(query_stats.execution_count) AS [Avg CPU Time],
query_stats.query_hash AS [Query Hash]
FROM (SELECT QS.*, SUBSTRING(ST.[text], (QS.statement_start_offset/2) + 1,
((CASE statement_end_offset
WHEN -1 THEN DATALENGTH(st.[text])
ELSE QS.statement_end_offset END
- QS.statement_start_offset)/2) + 1) AS statement_text
FROM sys.dm_exec_query_stats AS QS
CROSS APPLY sys.dm_exec_sql_text(QS.sql_handle) AS ST) AS query_stats
GROUP BY query_stats.query_hash
ORDER BY [Avg CPU Time] DESC;
You can change the ORDER BY clause on below query to monitor other resources:
SELECT TOP 10
(total_logical_reads/execution_count) AS avg_logical_reads,
(total_logical_writes/execution_count) AS avg_logical_writes,
(total_physical_reads/execution_count) AS avg_phys_reads,
(total_worker_time/execution_count) AS avg_cpu_over_head,
total_logical_reads, total_logical_writes, total_physical_reads,
total_worker_time, execution_count, total_elapsed_time AS Duration,
plan_generation_num AS num_recompiles,
statement_start_offset AS stmt_start_offset,
(SELECT SUBSTRING(text, statement_start_offset/2 + 1,
(CASE WHEN statement_end_offset = -1
THEN LEN(CONVERT(nvarchar(MAX),text)) * 2
ELSE statement_end_offset
END - statement_start_offset)/2)
FROM sys.dm_exec_sql_text(sql_handle)) AS query_text,
(SELECT query_plan FROM sys.dm_exec_query_plan(plan_handle)) AS query_plan
FROM sys.dm_exec_query_stats a
--JUST CHANGE THE ORDER BY TO GET THE OTHER RESOURCES
ORDER BY (total_logical_reads + total_logical_writes)/execution_count DESC
You may want to store the result of queries like the above for later analysis.